Download Derivation of the Paschen curve law ALPhA Laboratory Immersion

Derivation of the Paschen curve law
ALPhA Laboratory Immersion
Arturo Dominguez
July 3, 2014
1
Objective
If a voltage differential is supplied to a gas as shown in the setup (Figure 1), an
electric field is formed. If the electric field applied is strong enough, an avalanche
process (the Townsend avalanche) is started which leads to the breakdown of the
gas and the formation of plasma. This document describes the physics of this
process and derives the law (Paschen’s law) that predicts the voltage differential
that needs to be supplied in order to create the plasma.
2
Experimental Setup
In Figure 1 the experimental setup for the Paschen curve experiment is shown.
A pair of parallel plate electrodes are placed inside a vessel that contains a gas
which can be air but can also be a more pure gas, like He, Ar, Ne, etc. While the
geometry of the setup is irrelevant to the qualitative behavior of the breakdown
voltage, the simple 1D geometry results in a simpler comparison with theory.
The variables that can be controlled are: the pressure of the gas, p, the distance
between the electrodes, d, and the voltage between the electrodes, V , as well as
the gas contained in the vessel.
3
Qualitative description
As the voltage difference is applied, the electric field will accelerate any free
charges that exist. Free electrons exist in the system due to random events
from a variety of mechanisms including the triboelectric effect or through astronomical particles traversing the vessel and ionizing neutral particles. If an
electron can gain more than the ionizing energy of the gas, UI (approximately
14eV for Nitrogen), the electron can ionize the neutral particle and create a
new free electron and a free ion. The new free electron can repeat the process and create a chain reaction called the Townsend Avalanche. The ion is
accelerated towards the cathode and as it hits it there is a chance that it will
1
Figure 1: The standard DC discharge setup shows an electron ionizing a neutral
particle. When the process becomes self sustaining, the Townsend Avalanche is
initiated and a plasma is formed.
2
free an electron from the cathode. This process is called secondary emission
and is responsible for the production of electrons that can sustain the plasma.
Once a threshold condition is met, the secondary electrons suffice to begin the
Townsend Avalanche.
4
Derivation of Paschen’s Law
Quantitatively, the electron avalanche can be described using the rate of ionization per unit length, α. For example, if α = 2cm−1 then within a cm of
electron travel along the x-axis, there will, on average, be 2 collisions with neutral particles that result in ionizing events. This results in the following equation
describing the increase of electron current density, Γe (x):
dΓe (x) = Γe (x)αdx,
(1)
Γe (x) = Γe (0)eαx .
(2)
which integrates to:
Since there is no current leaving or entering the vessel except through the electrodes, and there is no charge accumulation, the continuity equation results in:
Γ(0) = Γ(d),
(3)
that is, the current density at the cathode (x = 0) is the same as that at the
anode (x = d). Assuming a single ion species, Equation 3 can be rewritten as:
Γe (0) + Γi (0) = Γe (d) + Γi (d)
(4)
Γi (0) = Γe (d) − Γe (0) + Γi (d).
(5)
Using the fact that there is no input of ions from the anode, Γi (d) = 0. Using
this constraint, and substituting Equation 2 into Equation 5 results in:
Γi (0) = Γe (0)(eαd − 1).
(6)
As an ion reaches the cathode, the probability of it releasing a secondary electron
is given by the coefficient γ, therefore:
Γe (0) = γΓi (0).
(7)
At the threshold point where the secondary emission sustains the plasma and
the Townsend Avalanche is formed, Equations 6 and 7 are balanced and Γi (0)
can be substituted, leading to:
Γe (0)
= Γe (0)(eαd − 1)
γ
1
= eαd − 1
γ
αd = ln (1 + 1/γ).
3
(8)
(9)
(10)
Figure 2: The volume associated with an individual neutral gas particle is given
by vol = 1/n = σλ.
Figure 3: The volume associated with an individual neutral gas particle is given
by vol = 1/n.
Equation 10 gives one constraint on the ionization coefficient.
While we’ve defined the role of α, we haven’s discussed a physical derivation of
what its value should be. In the following section we will tackle that.
As an electron is accelerated through the gas, the distance it travels, on
average, before it collides with a neutral particle is given by the mean free
path (mfp), λ. The volume occupied by a single neutral particle is given by
vol = 1/n, where n is the neutral density, as shown in Figure 2. The face of
the cylinder is the cross section of the collision between the neutral particle and
the electron, or σ ≈ π(re + rn )2 ≈ πrn2 where re and rn are the electron and
neutral particle’s radii respectively. Therefore, the volume occupied per neutral
is: vol = 1/n = σλ. Since the neutral gas follows the ideal gas law, p = nkB T ,
where p is the neutral pressure, T is the temperature of the neutrals and kB is
the Boltzmann constant, the mean free path can be rewritten as:
λ=
1
kB T
=
σn
σp
(11)
λ is the rate of collisions per length in the x-axis of electrons hitting neutral
particles. Note that if the electron is energetic enough, the collision will ionize
the particle, but this is not generally true.
If you follow the flow of electrons at a position x0 , one can define the current
density of free electrons, Γe free (x0 + ∆) which is the current density of free electrons from x0 that have reached x0 + ∆ without having collided with a neutral.
4
In Figure 3 it is observed that as the electrons travel across the length, ∆, some
will collide with neutrals and are lost from the Γe free . The rate of collisions per
unit length is, as was discussed earlier, λ. Therefore, the differential equation
that determines Γe free (x) is given as:
dΓe free (x)
=
Γe free (x0 + ∆)
=
dx
λ
Γe free (x0 )e−∆/λ
Γe free (x0 + ∆)
Γe free (x0 )
=
e−∆/λ
−Γe free (x)
(12)
(13)
(14)
Note that the RHS of Equation 14 is independent of x0 , hence, at any point in
x, the probability that a free electron has traversed a distance of at least ∆ is
given by:
P (distance traveled ≥ ∆) = e−∆/λ
(15)
Now, let’s get to α. If every collision between an electron and a neutral resulted
in an ionization, then α = 1/λ, but only the proportion of those electrons that
have enough energy to ionize the neutral particle, UI , will cause the ionization,
therefore:
P (electrons with energy ≥ UI )
α=
(16)
λ
Since the electrons gain energy by being accelerated down the electric field, E,
then we can define λI using:
UI
=
eEλI
(17)
UI
λI =
(18)
eE
UI d
λI =
(19)
eV
where E = V /d has been used. λI is, therefore, the distance that an electron
must travel in the electric field in order to gain the necessary energy, UI , to
ionize the neutral upon collision. Therefore, Equation 20 can be rewritten as:
P (distance traveled ≥ λI )
e−λI /λ
=
,
(20)
λ
λ
where Equation 15 has been used. Multiplying d on both sides of Equation 20,
we obtain:
α=
e−λI /λ
λ
d
ln (1 + 1/γ) =
e(−UI d/eV )/(kB T /σp)
kB T /σp
σ
ln (1 + 1/γ) =
(pd)e(−UI σ/ekB T )(pd/V ))
kB T
−Bpd
ln (ln (1 + 1/γ)) − ln (Apd) =
V
αd = d
5
(21)
(22)
(23)
(24)
Figure 4: Equation 25 using A = 1.5mT orr−1 m−1 , B = 36V /(mT orr × m) and
γ = 0.02
where A ≡ σ/kB T and B ≡ UI σ/ekB T and we have used Equations 10, 11 and
19. Finally, since V is really the voltage right at the point of breakdown, we
can change its notation to VBD (breakdown voltage) and put it on the LHS:
VBD =
Bpd
ln (Apd) − ln (ln (1 + 1/γ))
(25)
This is the final form of Paschen’s Law. If we take pd as the abscissa (x − axis)
and VBD as the ordinate (y − axis) then the range (as we will see) is pd =
(ln (1 + 1/γ)/A, ∞).
For A = 1.5mT orr−1 m−1 , B = 36V /(mT orr × m) and γ = 0.02 which
are typical values in air at room temperature using stainless steel electrodes,
Equation 25 leads to the plot shown in Figure 4.
Note the existence of a minimum in the curve. The pd and VBD at the minimum
BD
can be found using the fact that at the critical value, dV
d(pd) = 0:
dVBD
=
d(pd|min )
B(D − 1) =
D
=
pd|min
=
VBD min
=
BD −
1
pd|min (Bpd|min )
D2
=0
(26)
0
(27)
ln (ln (1 + 1/γ)) − ln (Apd|min ) = 1
ln (1 + 1/γ)
e
A
B ln (1 + 1/γ)
e
A
(28)
(29)
(30)
where D ≡ ln (ln (1 + 1/γ)) − ln (Apd|min ) is the denominator of Equation 25
and e in Equations 29 and 30 is Euler’s number.
6
Using Equations 10 and 11, Equation 29 can be rewritten as:
α|min =
e−1
λ
(31)
Or, comparing it to Equation 20, at the minimum: λI = λ. This makes sense
intuitively, it’s taking the electrons the mfp to accelerate just enough to ionize
the neutrals. If you were to increase λI , a lot of the collisions would not lead
to ionization, if you increase λ then the rate of collisions would drop. (from the
denominator in Equation 20).
We also see that when D = 0, VBD → ∞, this occurs when:
ln (ln (1 + 1/γ)) − ln (Apd|inf )
ln (1 + 1/γ)
α|inf
=
0
= Apd|inf
1
=
λ
(32)
(33)
(34)
This result also makes sense, since the rate of ionizing collisions can’t be greater
than the rate of collisions (1/λ). This sets the lower limit on pd as pd >
ln (1 + 1/γ)/A as discussed before.
5
Discussion of the minimum
The conditions for the minimum are shown in Equations 29 and 30. Its existence
can be understood by looking at the extremes: if pd is too big, since λ ∼ 1/p,
λ is very small, therefore the electrons will collide too much and won’t acquire
the ionizing energy UI necessary. On the other hand, if pd is too small, that is
λ is big compared to the distance between the electrodes d, the electrons will
collide with the anode before they have a chance of ionizing the gas.
There are many implications of Paschen’s law, e.g., when constructing a fluorescent light bulb, where the distance between the electrodes d, is set by the
application, the pressure is set so as to be close to the minimum in order to
minimize the necessary voltage needed, leading to lower power consumption.
The typical pressure inside a fluorescent bulb is ≈ 2T orr..
Another consequence of Paschen’s law is that if we’re dealing with atmospheric
plasmas (p ≈ 760T orr) the distance between electrodes for the minimum condition is calculated (using the same conditions as in Figure 4) to be ∼ 10µm.
This leads to the label of microplasmas for these type of discharges and their
consequential difficulty in attaining experimentally.
7